Quantum Metrology

With almost a century’s development, quantum mechanics has been walking into the field of technological science from the purely theoretical one. Quantum technology is the outcome of this trend and has shown its power in some tradi-tional high-tech field, such as communication and electronic industry. Quantum metrology, as a emerging part of quantum technology, also draw plenty of atten-tion in the community and showed its potential. In this paper, we will study the quantum metrology with the help of Cramer-Rao theory.In the first chapter, we briefly introduce the the development of quantum metrology and review the classical Cramer-Rao inequality. As the cornerstone of this paper, we give detailed derivations of quantum Cramer-Rao inequalities for both single- and multi-parameter estimations. All the derivations are based on the Cauchy-Schwartz inequality for matrices. From the quantum Cramer-Rao inequalities, one can see that for the single-parameter estimation, the variance of parameter under estimation is lower bounded by the inverse of quantum Fisher information; for the multi-parameter estimation, the counterpart lower bound is the inverse matrix of quantum Fisher information matrix. These facts imply that quantum Fisher information and quantum Fisher information matrix are very crucial indexes to quantify the precision of parameters that a system can provide.In the second chapter, we introduce the calculations of quantum Fisher in-formation and quantum Fisher information matrix in detail. Most of the calcula-tions given in the past were focus on the full rank density matrices. Here in this paper, we consider the fact that density matrix could be non-full rank. The rig-orous analytical expressions of quantum Fisher information and quantum Fisher information matrix are given. This expressions imply that quantum Fisher in-formation and quantum Fisher information matrix can be totally determined by the eigenvalues and eigenstates within the support of the density matrix. Next, we discuss the calculation of symmetric logarithmic derivative. Based on the Lyapunov representation of symmetric logarithmic derivative, we provide a new method to calculate it. Same as the Lyapunov representation, this method is independent of the specific representation of density matrices, and is particularly useful for some cases, such as the one in which the nth order anti-commutation be-tween the density matrix and its partial derivative is periodic or can be truncated. Utilizing this method, we successfully provide the representation-free analytical expression of symmetric logarithmic derivative and quantum Fisher information for any two-level system.Unitary parametrization is very common and useful in quantum metrology. It is the foundation for many metrological processes in optical and atomic in-terferometers. In the third chapter of this paper, we focus on the theory of unitary parametrization processes. We introduce a characteristic operator which only relies on the parametrization process and the parameter under estimation. With the help of this operator, the calculation of quantum Fisher information and quantum Fisher information matrix can be separated into two parts, first the calculation of the characteristic operator and then its matrix element in the eigenspace of the initial state. For the cases that the initial states are pure, the quantum Fisher information is just the variance of the characteristic operator on the initial state and the quantum Fisher information matrix is the covariance matrix. After giving the calculation procedure of the characteristic operator, we discuss two scenarios. The first one is with the exponential form initial state and the second one is a series of physical systems. For both scenarios, we give the analytical expression of the characteristic operator and the quantum Fisher information.In the fourth chapter, we discuss the quantum metrology in optical Mach-Zehnder interferometer. As the beginning, we briefly introduce the Mach-Zehnder interferometer and its theoretical representation. Then we consider a scenario, in which one import of the interferometer is an odd or even state and the other one is an arbitrary state. We provide the expression of quantum Fisher information in this scenario. From this expression, we find the phase-matching condition for the initial state to optimize the quantum Fisher information. Besides, we also discuss the phase-matching conditions for the cases with unbalanced beam split-ters, as well as photon losses in the interferometer. The multi-phase estimation in interferometer is also important and interesting. In the fourth chapter, we discuss the multi-phase estimation with a generalized entangled coherent state for both linear and nonlinear parametrization processes. Compared with the parameter precision given by the independent estimation with entangled coherent state, the generalized coherent state can provide a better precision, which is more insensi-tive to the phase number. This fact implies that the simultaneous estimation has a obvious advantage than the independent estimation for a large phase number case.In the fifth chapter, we summarize this paper and give some expectation on the development of quantum metrology in the future.